Subband indexing methods and systems

ABSTRACT

Methods and systems for indexing subband selections are disclosed. The selected subbands can be represented as a binary sequence and the index can be determined by employing a choose function, where a choose function value for a particular selected bit position is defined by a selection occurrence count of the selected bit position that is chosen from a bit position count of the selected bit position. In particular, the index can be the summation of choose function values for each selected bit position.

RELATED APPLICATION INFORMATION

This application claims priority to provisional application Ser. No.61/258,402 filed on Nov. 5, 2009 and provisional application Ser. No.61/261,552 filed on Nov. 16, 2009, each of which is incorporated hereinby reference.

This application is related to commonly assigned application Ser. No.12/940,761 and commonly assigned application Ser. No. 12/940,777, filedconcurrently herewith and incorporated herein by reference.

BACKGROUND

1. Technical Field

The present invention relates to indexing methods and systems and, inparticular, to methods and systems for indexing signal options employedin communication links.

2. Description of the Related Art

In many communication systems, control signals are employed betweentransmitters and receivers to enable the identification and proper setupof a communication link. These signals include, but are not limited to,reference signals and signals conveying information of other signaloptions such as modulation format, coding rate, precoding index,assigned subbands and so on. For example, certain communication systemsdynamically allocate a number of subbands from a set of availablesubbands to the users it services for purposes of transmitting data tothe users on the allocated subbands. It is often desirable to reduce thesignaling, as it increases overhead of the communication systems.Different indexing approaches for conveying control information havebeen described in the literature. For example, with regard to subbandindexing, a conventional approach employs a known algorithm that isbased on the enumerative source coding problem to compute indices for aset of binary permutations.

SUMMARY

In contrast to existing methods, exemplary embodiments described hereinprovide relatively simple and efficient means for indexing selectedsignal options and for determining selected signal options from indicesfor purposes of establishing a communication link. For example,exemplary implementations are directed to methods for transmittingcontrol information to permit the establishment of a communication link.In accordance with one method, a vector including one or more selectedelements representing one or more signal options selected from aplurality of available signal options can be obtained. The vector can berepresented as a sequence, which can be a binary sequence, and theelements of the vector can be represented as sequence positions or bitpositions. In addition, the signal options can be subbands on which thecommunication link can be established. Moreover, an index for the vectorcan be determined by employing the products of the value of each elementwith a value of a worth function for the element that is dependent on aselection occurrence count of the element and on an element positioncount of the element. Alternatively or additionally, the index can bedetermined as the summation of worth function values, where a particularworth function value is dependent on a selection occurrence count of aparticular selected sequence position and on a sequence position countof the particular selected sequence position. For example, the worthfunction can be a choose function that is defined by a selectionoccurrence count of a particular selected bit position that is chosenfrom a bit position count of the particular selected bit position.Alternatively or additionally, the index can be a summation of worthfunction values for each selected element that are dependent on anelement position count of the respective selected element, where thesummation is unique for any permutation of a selected element set. Inone example, the index can be determined by computing

${{I(S)} = {\sum\limits_{k = 1}^{M}\begin{pmatrix}P_{k} \\k\end{pmatrix}}},$wherein I(S) is the index, S is a binary sequence denoting selectedsubbands, M is the total number of selected subbands, k is a selectionoccurrence count of a selected subband and P_(k) is a subband positioncount in the binary sequence for a selected subband corresponding to k.Further, the index can be transmitted to enable the establishment of acommunication link in accordance with the selected signal options.

Other implementations are directed to methods for receiving controlinformation to permit the establishment of a communication link. Here,an index that is an indication of a vector including one or moreselected elements representing one or more signal options selected froma plurality of available signal options can be received. Similar to themethods for transmitting control information, the vector can berepresented as a sequence, which can be a binary sequence, and theelements of the vector can be represented as sequence positions.Moreover, the index can be dependent upon the products of the value ofeach element with a value of a worth function for the element that isdependent on a selection occurrence count of the element and on anelement position count of the element. Alternatively or additionally,the index can be a summation of worth function values, where aparticular worth function value is dependent on a selection occurrencecount of a particular selected sequence position and a sequence positioncount of the particular selected sequence position. In addition, thevector can be determined from the index and data signals can betransmitted or received in accordance with the one or more selectedsignals options.

An alternative implementation is directed to a transmitter system. Thesystem can include an indexing module that is configured to determine anindex for a binary sequence including one or more selected bit positionsrepresenting one or more subbands selected from a plurality of availablesubbands. The index can be a summation of choose function values, whereany particular choose function is defined by a selection occurrencecount of a particular selected bit position that is chosen from a bitposition count of the particular selected bit position. The system canfurther include a transmitter that is configured to transmit the indexto enable the establishment of a communication link on the one or moreselected subbands.

An alternative implementation is directed to a receiver system. Thesystem can include a receiver that is configured to receive an indexthat is an indication of a sequence including one or more sequencepositions representing one or more signal options selected from aplurality of available signal options. The index can be a summation ofworth function values, where a particular worth function value isdependent on a selection occurrence count of a particular selectedsequence position and a sequence position count of the particularselected sequence position. The system can further include a decodingmodule that is configured to determine the sequence from the index topermit the establishment of a communication link in accordance with theone or more selected signal options.

These and other features and advantages will become apparent from thefollowing detailed description of illustrative embodiments thereof,which is to be read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF DRAWINGS

The disclosure will provide details in the following description ofpreferred embodiments with reference to the following figures wherein:

FIG. 1 is a block/flow diagram of an exemplary communication system.

FIG. 2 is a block/flow diagram of an exemplary method for establishing acommunication link by employing indexing and reverse indexingprocedures.

FIG. 3 is a block/flow diagram of an exemplary reverse indexing method.

FIG. 4 is a block/flow diagram of an exemplary reverse indexing methodthat sequentially detects selected sequence positions.

FIG. 5 is a block/flow diagram of an exemplary reverse indexing methodthat employs characteristics of the choose function to simplifyoperations.

FIGS. 6 and 7 are block/flow diagrams of exemplary reverse indexingmethods that detect selected sequence positions by evaluating sequencepositions in a non-sequential order.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Systems and methods that permit efficient encoding and decoding ofvarious control signals for the identification and proper setup of acommunication link are disclosed. Such control signals can be utilizedin wired and/or wireless systems and can include information indicatingmodulation format, coding rate, precoding index, assigned subbands, etc.It should also be noted that the approaches described herein meet thecurrent orthogonal frequency division multiplexing (OFDM) basedstandards, such as the Institute of Electrical and Electronics Engineers(IEEE) 802.16m standard or Long Term Evolution (LTE) advanced standard,where a subset of tones are selected, properly indexed and then signaledto the receiver.

Exemplary embodiments described herein below provide simple andefficient means for indexing and reversing indexing a sequence ofselected signal options. In particular, the indexing and reverseindexing procedures provide several benefits that render themsubstantially more efficient than other procedures currently utilized inthe art. For example, the indices can be configured so that they areconsecutive integers denoting each possible variation of signal optionselections. Moreover, if the selected options are configured as binarysequences and sorted numerically in a list, the order of the indexparallels the numerical order of the sequences. In addition, a closedform expression can be employed to calculate an index for any sequencedenoting signal options or to determine the sequence from the index.Furthermore, the computational complexities of calculating the index forany sequence and the reverse procedure of calculating the sequence fromits index are very low. One way in which these benefits are achieved isby employing worth function values to determine indices that aredependent on both the position of a selected element in the sequence andthe selection occurrence count of that element value, as discussedfurther herein below.

Referring now in detail to the figures in which like numerals representthe same or similar elements and initially to FIG. 1, an exemplarycommunication system 100 in which various method embodiments describedherein can be implemented is illustrated. The system 100 can include atransmitter 110 and a receiver 150, which may communicate viacommunication links 119 and 120. As indicated above, the communicationlinks 119 and 120 can be wired or wireless. For example, in the wirelessscenario, the transmitter 110 can be implemented as a base station in anOFDM system. Alternatively, in the wired scenario, the transmitter 110can be embodied as an optical terminal that assigns subbands for anoptical communication link, which can be implemented as link 120.Furthermore, it should be noted that although only one receiver is shownfor purposes of brevity, the system 100 can include a plurality ofreceivers. Here, the transmitter 110 may comprise a controller 112 thatis configured to manage operations at the transmitter. In addition, thecontroller 112 can be configured to determine information fortransmission to the receiver that enables the proper establishment ofthe communication link 120. Here, the communication link 120 can be usedto transmit data, such as, for example, broadcast or multicast data. Asstated above, the determined information can comprise selected signaloptions, including an assigned modulation format, coding rate, precodingindex, subbands, etc. In particular, the controller 112 can employ anindexing module 114 to determine an appropriate index that indicates theinformation which enables the receiver 150 to establish thecommunication link 120. Operations of the indexing module 114 arediscussed in more detail herein below with respect to methodembodiments. Further, the controller 112 and the indexing module 114 canutilize a storage medium 113 to aid in determining indices indicatingselected signal options. For example, an optional lookup table 115 canbe stored in the storage medium 113 and can be employed by the indexingmodule 114 to determine the indices, as discussed in more detail hereinbelow. In addition, the transmitter system 110 can include a transmitter116 that can be configured to transmit information to the receiver 120.For example, an initial communication link 119 can be employed totransmit the determined index over, for example, a subband or controlchannel dedicated for configuration information.

In turn, the receiver system 150 can include a receiver 156 that can beconfigured to process received signals and decode informationtransmitted by the transmitter 110. For example, the receiver 156 caninclude a demodulator and a decoding module. Further, the receiversystem 150 can include a processor 152 that can be configured to controloperations of the receiver system 150. In particular, the processor 152can employ a decoding module 154 to decode the index received from thetransmitter 110 and thereby determine selected signal options. Asindicated above, the selected signal options can be utilized by thereceiver 150 to establish the communication link 120. Operations of thedecoding module 154 are also discussed in more detail herein below withrespect to method embodiments. Further, the processor 152 and theindexing module 114 can utilize a storage medium 153 to decode receivedindices and thereby determine any corresponding selected signal options.Similar to the indexing module 114 of the transmitter 110, the decodingmodule 154 can employ an optional lookup table 155 stored in the storagemedium 153 to determine the selected signal options, as discussed inmore detail herein below.

It should be understood that embodiments described herein may beentirely hardware, entirely software or including both hardware andsoftware elements. In a preferred embodiment, the present invention isimplemented in hardware and software, which includes but is not limitedto firmware, resident software, microcode, etc.

Embodiments may include a computer program product accessible from acomputer-usable or computer-readable medium providing program code foruse by or in connection with a computer or any instruction executionsystem. A computer-usable or computer readable medium may include anyapparatus that stores, communicates, propagates, or transports theprogram for use by or in connection with the instruction executionsystem, apparatus, or device. The medium can be magnetic, optical,electronic, electromagnetic, infrared, or semiconductor system (orapparatus or device) or a propagation medium. The medium may include acomputer-readable storage medium such as a semiconductor or solid statememory, magnetic tape, a removable computer diskette, a random accessmemory (RAM), a read-only memory (ROM), a rigid magnetic disk and anoptical disk, etc.

A data processing system suitable for storing and/or executing programcode may include at least one processor coupled directly or indirectlyto memory elements through a system bus. The memory elements can includelocal memory employed during actual execution of the program code, bulkstorage, and cache memories which provide temporary storage of at leastsome program code to reduce the number of times code is retrieved frombulk storage during execution. Input/output or I/O devices (includingbut not limited to keyboards, displays, pointing devices, etc.) may becoupled to the system either directly or through intervening I/Ocontrollers.

Network adapters may also be coupled to the system to enable the dataprocessing system to become coupled to other data processing systems orremote printers or storage devices through intervening private or publicnetworks. Modems, cable modem and Ethernet cards are just a few of thecurrently available types of network adapters.

To illustrate aspects of exemplary embodiments, a set V of vectorsv_(i), i=1, . . . , K are considered. Here, elements of the vectors aretaken from a finite set of size b. For example, the set can be denotedby F, where F={0, 1, . . . , b−1}. In accordance with one exemplaryaspect, an index denoting a vector can be found as the summation of theproduct of the value of each element with its corresponding ‘worthfunction.’ The worth function here is a function of both the position ofthat element in the vector and the occurrence count of that elementvalue from the beginning of the vector. This procedure is different fromthe conventional approach mentioned above, in which the ‘worth function’is just a function of the position of the element in the vector.

For example, a case in which b=2 and a vector v₁=[100101] is considered,where there are ones in positions 0, 2 and 5 of the vector. The positionindex here is counted from zero for the rightmost element of the vector.However, as discussed further herein below, it should be noted that theparticular ordering of the elements of the vector is not important. Forexample, the indexing may start from 1 to N and may also start from theleftmost element position of the vector. It can be assumed that theposition of ones are indicated as P₁<P₂< . . . <P_(M), where in theabove-described example P₁=0, P₂=2, and P₃=5. As stated above, the worthfunction is a function of the occupancy count k and the actual positionP_(k) rather than just a function of P_(k) in positional numeralsystems. Thus, a worth function in accordance with exemplary aspects isin the form ƒ(k, P_(k)).

An example of such a worth function is

${f\left( {k,P_{k}} \right)} = {\begin{pmatrix}P_{k} \\k\end{pmatrix}.}$This function, for example, can be used to index all possiblepermutations of a string of length N with M ones and N−M zeros. It turnsout that the corresponding index orders all possible permutations inlexicographical order.

One exemplary application in which the indexing procedure can beemployed is to index subbands assigned to a receiver for use incommunicating with a transmitter. It should be understood, however, thatthis application is only used as an example. The index can denote anyone or more selected signal options from a plurality of available signaloptions for the communication link between the transmitter and thereceiver. Such signal options can comprise, for example, one or more ofmodulation format, coding rate, precoding index, assigned subbands, etc.Furthermore, any of these types can be combined in an index such that aparticular index can indicate several different selection options. Forexample, the index can indicate both a modulation and coding scheme(MCS), as understood in the art.

In this example, signal option selections can be denoted by a binarysequence of length N where only M positions corresponding to theselected options are one and the rest are zero. For example, with regardto subband signal options, each position in the sequence can correspondto a different subband, where selected subbands to be used by atransmitter and a receiver on a communication link can be denoted byones and other subbands not used for the link can be denoted by zeros.Such sequences are referred to as SPS (Subband Positions Sequence). Inthis example, there will be only

$\quad\begin{pmatrix}N \\M\end{pmatrix}$possibilities for SPS out of all 2^(N) possible binary sequences.

It should be understood that a general sequence S is an example of onevector in the set of V vectors mentioned above and can be generallyreferred to for the various signal options mentioned above. Thus, anyreference herein to a sequence position can correspond to an elementposition of a vector described above. Furthermore, the sequences neednot be binary.

The indexing procedure described above provides a simple and efficientway to index such general sequences. For example, in the case where theworth function is given by

${{f\left( {k,P_{k}} \right)} = \begin{pmatrix}P_{k} \\k\end{pmatrix}},$the index are efficient in the sense that they are exactly between 0 and

$\begin{pmatrix}N \\M\end{pmatrix} - 1.$Further, if binary sequences are treated as the binary number and sortednumerically, the index shows the order of the sequence in the sortedlist. Accordingly, indices for permutations of selected sequencepositions are arrangeable in lexicographical order corresponding to anumerical order of the permutations. In addition, for any sequence,there is a closed form expression to calculate the corresponding index.Moreover, the computational complexities of calculating the index forany sequence and the reverse procedure of calculating the sequence fromits index are very low.

It should be noted that the indexing procedure described above can beperformed on a permuted vector v′=Πv rather than the original vector v.In other words, it is possible to perform the same indexing procedure ona re-ordered vector v. A simple example of such a permutation is onethat completely reverses the position of the elements of the vector. Forexample, if the elements of the vector (of size 1×b) are indexed from 1to b, then the k^(th) element will be exchanged with the (b−k+1)^(th)element. Thus, it is equivalent to index the vector from N to 1 and theoccurrence of each element is counted from the last element of thevector instead of the first element of the vector.

Referring now to FIG. 2 with continuing reference to FIG. 1, a method250 for conveying indexing information to a receiver in accordance withan exemplary embodiment is illustrated. The method 200 may begin at step251, in which the indexing module 114 can obtain a sequence including atleast one selected sequence position representing at least one selectedsignal option from a plurality of available signal options. As statedabove, the signal options can comprise at least one of modulationformat, coding rate, precoding index or assigned subbands, in additionto other signal options. Furthermore, as indicated above, the sequencecan be binary and can have selected sequence or bit positions denoted byones or zeros. Embodiments in which the selected sequence or bitpositions are denoted by zeros are described in more detail hereinbelow. It should be noted that the sequence can be obtained at step 251in a variety of ways. For example, the controller 112, which can beconfigured to manage communication links with a plurality of receiverdevices, can determine the selected signal options for each of thecommunication links in accordance with known methods. Here, the indexingmodule 114 can receive sequences denoting selected bit positions foreach of the receivers served by the transmitter 110 from the controller112. Alternatively, the indexing module 114 can obtain the sequence atstep 251 by determining the sequence itself in a similar manner.

At step 252, the indexing module 114 can determine an index for asequence S. As indicated above, an index denoting a vector representinga sequence can be found as the summation of the product of the value ofeach element with its corresponding ‘worth function’:

${I(S)} = {\sum\limits_{k = 1}^{M}\;{{f\left( {k,P_{k}} \right)}.}}$Further, as stated above, where the selected sequence positions in abinary sequence are denoted by ones, the summation

$\sum\limits_{k = 1}^{M}\;{f\left( {k,P_{k}} \right)}$can correspond to a summation of the product of the value of eachelement of a vector with its corresponding worth function ƒ(k, P_(k)).Here, such a binary sequence can denote a 1×N vector with M selectedelements, wherein the index is determined by summing the products of thevalue of each element with a value of a worth function for the element.Continuing with the example described above, let P_(k) denote the k^(th)nonzero position in the sequence where P_(M)> . . .>P_(k+1)>P_(k)>P_(k−1)> . . . >P₁. Thus, any particular value of theworth function ƒ(k, P_(k)) can be dependent on a selection occurrencecount, for example k, of a particular selected sequence position and ona sequence position count, for example P_(k), for the particularselected sequence position at P_(k). Here, it is assumed that thepositions are indexed from zero for the least significant position toN−1 for the most significant position. However, the indexing of thepositions can be from 1 to N and can be varied, as discussed above withregard to vector permutations. As stated above, the worth function canbe a choose function

${{f\left( {k,P_{k}} \right)} = \begin{pmatrix}P_{k} \\k\end{pmatrix}},$where k is chosen from P_(k). It should be noted that

$\quad\begin{pmatrix}P_{k} \\k\end{pmatrix}$is defined to be zero if and only if P_(k)<k, which signifies that it isnot possible to choose k elements out of P_(k) elements.

It should also be noted that although the indexing procedure has beengenerally described as finding an index as the summation of the productof the value of each element of a vector with its corresponding ‘worthfunction’ value, indexing procedures in accordance with exemplaryembodiments need not be limited in such a manner. For example, if theelements of a vector denoting a sequence employed ones to denoteselected positions of the sequence, then

${I(S)} = {\sum\limits_{k = 1}^{M}\;{f\left( {k,P_{k}} \right)}}$can be viewed as the summation of a product of the value of each elementof the vector with its worth function ƒ(k, P_(k)). However, the selectedor assigned positions can alternatively be denoted by zeros and theother positions can be denoted by ones. In this scenario in which zerosdenote the selected positions,

${I(S)} = {\sum\limits_{k = 1}^{M}\;{f\left( {k,P_{k}} \right)}}$can still hold, where P_(k) can denote the k^(th) zero position in thesequence and where P_(M)> . . . >P_(k+1)>P_(k)>P_(k−1)> . . . >P₁.Furthermore, in the case where the worth function is given by

${{f\left( {k,P_{k}} \right)} = \begin{pmatrix}P_{k} \\k\end{pmatrix}},$the index are also efficient in the sense that they are exactly between0 and

$\begin{pmatrix}N \\M\end{pmatrix} - 1.$Alternatively, the indexing procedure can remain the same, where P_(k)can denote the k^(th) nonzero positions in the sequence and where P_(M)>. . . >P_(k+1)>P_(k)>P_(k−1)> . . . >P₁. However, here, in alternativeembodiments, the receiver can simply assume that the zero positions inthe sequence indicate the selected or assigned positions for thereceiver.

At step 254, the controller 112 of the transmitter system 110 can directthe transmitter 116 to transmit the determined index information, whichcan be denoted as I_(M), to the receiver system 150 through, forexample, the communication link 119. As indicated above, the index canbe employed by the receiver system 150 to permit the establishment ofthe communication link 120 in accordance with the selected signaloptions. Optionally, the controller 112 can transmit an indication ofthe total number (M) of elements selected from the N elements of thesequence S, which in turn indicates the total number of selected signaloptions. For example, a case in which M can be in the range of 2 to 6 isconsidered. In such a case, three more bits can be added to signal thechoice of M. However, it is possible to add only one more bit and stillbe able to permit the receiver to determine the correct value for M. Forexample, the controller 112 can use an overflow bit at the position N ontop of the regular positions from 0 to N−1. In turn, the receiver 150,at step 258, described below, can initiate the reverse indexingprocedure by assuming the largest M, i.e., M=6. If the receiver 150determines that the most significant bit at position N is not set, thenthe receiver 150 can determine that M=6. If the receiver 150 determinesthat the most significant bit at position N is set to one, the receiver150 can determine that M≦5. Here, the value

$\begin{pmatrix}N \\M\end{pmatrix} = \begin{pmatrix}N \\6\end{pmatrix}$for the current M=6 should be deducted from the index to generate a newindex and it is determined that the value of M is less than or equal to5, M≦5. The receiver 150 can then repeat the procedure to test anddetermine if M=5 or M<5 in the same manner until all choices of M areexhausted. For example, if M=4, then the receiver 150 will determinethat the overflow bit at position N is zero, indicating that fourelements are selected from the N elements of the sequence S. In thegeneration of the index by the transmitter in 110 for the same examplewhen multiple choices for M exist, for example M=1, 2, . . . , 6, theindexing procedure is as follows. For M=6, i.e., the largest value, theindices are from 0 to

$\begin{pmatrix}N \\6\end{pmatrix} - 1.$For M=5, the indices are then calculated using

${{I(S)} = {\sum\limits_{k = 1}^{M}\;{f\left( {k,P_{k}} \right)}}},$as described above, but then the index is added with

$\begin{pmatrix}N \\6\end{pmatrix},$which in turn means

${I(S)} = {\begin{pmatrix}N \\6\end{pmatrix} + {\sum\limits_{k = 1}^{M}{{f\left( {k,P_{k}} \right)}.}}}$If M=4, then the same procedure using

${I(S)} = {\sum\limits_{k = 1}^{M}\;{f\left( {k,P_{k}} \right)}}$as described above is applied to find the index but then it is addedwith

${\begin{pmatrix}N \\6\end{pmatrix} + \begin{pmatrix}N \\5\end{pmatrix}},$which means

${I(S)} = {\begin{pmatrix}N \\6\end{pmatrix} + \begin{pmatrix}N \\5\end{pmatrix} + {\sum\limits_{k = 1}^{M}{{f\left( {k,P_{k}} \right)}.}}}$In general for any M between 1 and M_(max) the index is calculated as

${I(S)} = {{\sum\limits_{L = {M + 1}}^{M_{\max}}\begin{pmatrix}N \\L\end{pmatrix}} + {\sum\limits_{k = 1}^{M}{{f\left( {k,P_{k}} \right)}.}}}$

At step 256, the processor 152 of the receiver system 150 can receivethe determined index information I_(M) through the receiver 156 using,for example, the communication link 119.

At optional step 258, the processor 152 of the receiver system 150 candetermine the number (M) of elements selected from the N elements of thesequence S. For example, the processor can determine M as discussedabove with respect to step 254.

At step 260, the receiver can calculate the selected positions P_(k) foreach value of k in accordance with one or more of a variety of ways,discussed in more detail herein below.

At step 262, the transmitter 116 can transmit or receive data inaccordance with the selected signal options at the direction of thecontroller 112. For example, if the index denotes selected subbands,then data signals can be transmitted or received on the selectedsubbands in the communication link 120. Similarly, if a selected signaloption corresponds to a selected modulation or coding scheme, then thetransmitter can transmit or receive data signals on the communicationlink 120 using the selected modulation and coding scheme tomodulate/encode or demodulate/decode the data signals, respectively.

Similarly, at step 264, the receiver 156 can receive or transmit datasignals in accordance with the selected signal options at the directionof the processor 152. The communication link 120 can be established andutilized as discussed above with respect to step 262.

Returning to step 260 and referring to FIG. 3, with continuing referenceto FIGS. 1 and 2, a method 350 for determining the sequence S from thereceived index I_(M) that can be performed to implement step 260 isillustrated. Further, Algorithm 1, provided in Table 1 below, can alsobe performed by the decoding module 154 of the receiver system 150 toimplement the method 350. The worth function ƒ(k, x) can, for example,be the choose function

$\begin{pmatrix}x \\k\end{pmatrix}.$

TABLE 1 Algorithm 1 I_(M) = I for each k = M,M − 1,...2 do Find thelargest value for x such that I_(k) − f(k,x)≧0 I_(k−1) = I_(k) − f(k,x)P_(k) = x end P₁ = I₁

At step 352, the decoding module 154 of the receiver system 150 caninitialize the process and can set I_(M) to I and k to M.

At step 354, the decoding module 154 can determine whether k has reachedthe value of 1. If not, then the method 350 can proceed to step 356, inwhich the decoding module 154 can find the largest value for x such thatI_(k)−ƒ(k, x)≧0. As stated above, the choose function

$\begin{pmatrix}x \\k\end{pmatrix}\quad$can, for example, be employed as the worth function ƒ(k, x).

At step 358, the decoding module 154 can designate, as a given selectedsequence position, the sequence position with the largest sequenceposition count that renders a given worth function value, for aselection occurrence count denoting the given selected sequenceposition, less than or equal to an index value. For example, here, forthe selected bit or sequence position corresponding to the selectionoccurrence count k, the decoding module 154 can designate, as theselected bit or sequence position, the sequence position with thelargest sequence position count x that renders a given worth functionvalue ƒ(k, x) less than or equal to the index value I_(k). Accordingly,the decoding module 154 can set P_(k) to x. In addition, the decodingmodule 154 may also set I_(k−1) to I_(k)−ƒ(k, x), and k to k−1. Thus,while the index I_(k) can be initially set to I_(M), the index I_(k) canbe reset as the difference between the given worth function value ƒ(k,P_(k)) and the index I_(M) through a subtraction operation. Thereafter,the method 350 can proceed to step 354 and can be iterated until k isset to 1. In particular, the designating and the subtraction at step 358can be iterated for each selected sequence position by utilizing a valueof the index resulting from a preceding subtraction iteration. Forexample, the subtraction, I_(k−1)=I_(k)−ƒ(k, x), performed afterdetermining the selected sequence position for count k is employed asthe index value for determining the following selected sequence positionfor count k−1. As such, the index value I_(k) is effectively thedifference between the received index I_(M) and the sum of worthfunction values ƒ(j, P_(j)) for j>k.

If the decoding module 154 determines that k is set to 1 at step 354,then the method may proceed to step 360, in which the decoding module154 sets P₁ to I₁, at which point all of the selected signal optionshave been found.

Step 260 and the method 350 can be implemented in a variety of ways.Different exemplary alternatives for implementing step 260 and themethod 350 are described herein below with respect to FIGS. 4-6.

Referring now to FIG. 4, with continuing reference to FIGS. 1-3, amethod 450 for determining the sequence S from the received index I_(M)that can be performed to implement step 260 and method 350 isillustrated. In accordance with the method 450, the decoding module 154can sequentially evaluate sequence or bit positions to determine whichof the positions are nonzero or selected positions. Performing thesequential detection of the nonzero or selected positions from the mostsignificant position to the least significant position can takeadvantage of several properties. For example, when the most significantnonzero or selected position is found, the next nonzero or selectedposition would only be to the right of it. Thus, the decoding module 154can evaluate only positions that sequentially follow positions that havebeen found to determine any remaining selected sequence positions.Moreover, it is sufficient to examine positions N, N−1, . . . 0 todetermine the selected positions and then terminate the process when allM selected positions are found.

Algorithm 2 provided in Table 2 below can be employed by the decodingmodule 154 to implement the method 450. The worth function ƒ(k, j) can,for example, be the choose function

$\begin{pmatrix}j \\k\end{pmatrix}.$

TABLE 2 Algorithm 2: Sequential Detection Based Procedure k = M; foreach j = (N−1):−1:0 do if I ≧ f(k,j) then P_(k) = j; I = I − f(k,j);k=k−1; end end

At step 452, the decoding module 154 of the receiver system 150 caninitialize the process and can set k to M, j to N−1, and I to I_(M).

At step 454, the decoding module can determine whether j is zero. Ifnot, then the method can proceed to step 456 in which the decodingmodule 154 can determine whether I is greater than or equal to ƒ(k, j).As stated above, the choose function

$\begin{pmatrix}j \\k\end{pmatrix}\quad$can, for example, be employed as the worth function ƒ(k, j).

If I is less than ƒ(k, j), then the method may proceed to step 458, inwhich the decoding module 154 can decrement j by one. Thereafter, themethod may proceed to step 454 and may be repeated.

If I is greater than of equal to ƒ(k, j), then the method may proceed tostep 460, in which the decoding module 154 can set P_(k) to j, I toI−ƒ(k, j), and k to k−1. Thereafter, the method may proceed to step 458and can be repeated.

Returning to step 454, if the decoding module 154 determines that j hasreached zero, then the method may end.

If the choose function is employed as the worth function, then themethod 450 can be modified to reduce the complexity of the process byutilizing particular properties of the binomial (choose) function. Forexample, referring now to FIG. 5, with continuing reference to FIGS.1-3, a method 550 for determining the sequence S from the received indexI_(M) that can be performed to implement step 260 and method 350 isillustrated. Similar to the method 450, the nonzero or selected bits canbe determined sequentially. However, here, properties of the choosefunction can be exploited, as threshold values for the comparison of thechoose function value with I could be recursively found from theprevious threshold value by employing only one multiplication and onedivision operation. The operations for finding the threshold values aresimilar to those discussed further herein below with respect to the useof a reduced lookup table. Accordingly, only when the process is startedfor the new nonzero position is the choose operation performed directly.Considering the fact that the last position can be found directlywithout performing any choose operation, the total number of times thata choose operation is performed would be M−1. Algorithm 3 provided inTable 3 below can be employed by the decoding module 154 to implementthe method 550.

TABLE 3 Algorithm 3: Reduced Complexity Sequential Detection Procedurefor Choose Functions   k = M ; FL = 0 ; for each j = (N − 1) : −1:0 do if FL == 0 then     ${{Threshold}\; = \;\begin{pmatrix}j \\k\end{pmatrix}};$    else    ${{Threshold}\; = \;{{Threshold}\;*\;\frac{\left( {j - k + 1} \right)}{\left( {j + 1} \right)}}};$ end  if I ≧ Threshold then   P_(k) = j ;    ${I = {I - \begin{pmatrix}j \\k\end{pmatrix}}};$   k=k−1   if k == 0 then     P₁= I     Break;   end  FL = 0;  else   FL = 1;  end end

The method 550 can begin at step 552, in which the decoding module 154can set variables k to M, FL to zero, j to N−1 and I to the receivedindex I_(M).

At step 556, the decoding module 154 can determine whether the variableFL is zero. Here, FL is used to permit the decoding module 154 todetermine whether a threshold for comparison with the index should becomputed with a choose function operation or through simpler operations,as discussed further herein below.

If the decoding module 154 determines at step 556 that the variable FLis zero, then the method can proceed to step 558, in which the decodingmodule 154 can set the threshold to

$\begin{pmatrix}j \\k\end{pmatrix}.$If the decoding module 154 determines at step 556 that the variable FLis not zero, then the method can proceed to step 560, in which thedecoding module 154 can set the threshold to the product of the value ofthe preceding threshold with

$\frac{\left( {j - k + 1} \right)}{\left( {j + 1} \right)}.$Here, step 560 permits a recursive determination of the threshold forcomparison with an index value for a particular bit position count j byusing properties of the choose function. Specifically, the decodingmodule 154 can apply simple sum, subtraction and division operations asopposed to a choose function operation to determine the threshold.

After step 558 or 560, the method may proceed to step 562 in which thedecoding module 154 can determine whether the current index value isgreater than or equal to the threshold. If the current index value isless than the threshold, then the method may proceed to step 564, inwhich the decoding module 154 can set the variable FL to one, indicatingthat the simpler operations should be performed to determine the nextthreshold in the following iteration. Accordingly, the recursivecomputation of steps 562, 564, 566 and 560 can be performed until theselected sequence position for k has been found (i.e. until I is greaterthan or equal to the threshold in this example). In particular, thedecoding module 154 can execute operations other than a choose functionoperation to perform the recursive computation, thereby reducingcomplexity of the process. It should be noted that, in effect, the valueof the threshold determined at step 560 for a particular sequenceposition count j and for a particular selection occurrence count k isequal to the choose function value

$\begin{pmatrix}j \\k\end{pmatrix}{\quad.}$Thus, comparison of the index value I with the threshold can essentiallyimplement the determination of whether a given choose function value isless than or equal to a corresponding index value, as discussed abovewith respect to step 356 of the method 350.

If the current index value is greater than or equal to the threshold,then the method may proceed to step 568, in which the decoding module154 can set the variable P_(k) to j, can decrement I by the choosefunction value

$\begin{pmatrix}j \\k\end{pmatrix}\quad$and can decrement k by one. The method can proceed to step 570, in whichthe decoding module 154 can determine whether k is set to 1. If k is setto one, then the method can proceed to step 574 in which P₁ is set tothe current index value. Thereafter, the method may end, as all valuesof P_(k) have been determined. If, at step 570, the decoding module 154determines that k is not one, then the method may proceed to step 572,in which the decoding module 154 can set the variable FL to zero. Here,an FL value of zero indicates that a choose operation should beperformed to determine the next threshold in the following iteration.

After step 572 or 564, the method may proceed to step 566, in which thedecoding module 154 can decrement the value of the variable j by one.Thereafter, the method can proceed to step 556 and can be repeated untilP(1) is found.

In accordance with other exemplary embodiments, as opposed tosequentially searching for nonzero or selected positions, the selectedpositions can be searched for and found in a different order. Forexample, the first nonzero or selected position is known to be somewherein the position M−1 up to N−1. Here, the evaluation of sequencepositions to determine selected sequence positions can begin from themiddle of this range such that the comparison with the index isperformed with respect to the bit position at the middle of the range.Thereafter, the process can be iterated such that, at each step, thedecoding module 154 can divide the range in half until the position ofthe first nonzero or selected bit is found. When the selected bit isfound, then the selected bit can be designated as P(M). The decodingmodule 154 may then update the residual index I_(M-1), as discussedabove with respect to Algorithm 1 and FIG. 3. Subsequently, the secondnonzero or selected position would be somewhere in the range of M−2 toP(M). The same procedure discussed above with respect to the firstselected or nonzero position can be employed to find the second nonzeroposition. Similar to the first selected position, the second selectedposition can be designated as P(M−1) and the residual index I_(M-2) canbe updated in the same manner discussed above with regard to algorithm 1and FIG. 3. At this point, the third nonzero position would be in therange of M−3 to P(M−1). The same procedure can be repeated until thelast nonzero position is found. For the last selected bit, no additionalcomputation is necessary, as the residual index I₁ directly points tothe position of the selected bit. Accordingly, the selected sequencepositions can be evaluated and found in a non-sequential order.Algorithm 4 of Table 4, provided below, illustrates one exemplaryimplementation of searching for selected positions in a non-sequentialorder. It should be emphasized that, as discussed above, the worthfunction ƒ(k, j) can be the choose function

$\begin{pmatrix}j \\k\end{pmatrix}{\quad.}$

TABLE 4 Algorithm 4   P = zeros(M + 1); P(M + 1) = N; for each k = M :−1:2 do  up = P(k + 1)−1;  lo = k − 1;  while up ≠ lo do   ${j = {{lo} + {{ceil}\left( \frac{{up} - {lo}}{2} \right)}}};{{{if}\mspace{14mu} I}\; \geq {f\mspace{11mu}\left( {k,j} \right)\mspace{14mu}{then}}}$   lo = j;    else    up = j − 1;   end   P(k) = lo   I = I − f(k,P(k))end P(1)=1

Referring now to FIGS. 6 and 7 with continuing reference to FIGS. 1-3,methods 600 and 650 for determining the sequence S from the receivedindex I_(M) that can be performed to implement step 260 and method 350are respectively illustrated. In particular, methods 600 and 650 searchfor selected bit positions in a non-sequential order. In accordance withexemplary embodiments, Algorithm 4 can be performed by the decodingmodule 154 to implement method 600 and/or method 650.

Method 600 can begin at step 602, in which the decoding module 154 candetermine a window of sequence position counts between which aparticular selected sequence position P_(k) is located. For example,with reference to method 650, illustrating a more specificimplementation of the method 600, the decoding module 154 can set thevariable P(M+1) to N and can set the variable k to M at step 652.Thereafter, the method 650 can proceed to step 654, in which thedecoding module 154 can determine whether k is set to 1. If k is not setto 1, then the method 650 can proceed to step 656, in which the decodingmodule 154 can determine the window of bit position counts between whicha particular selected bit position P_(k) is located. Specifically, atstep 656, the decoding module 154 can set the upper threshold of thewindow to P(k+1)−1 and the lower threshold of the window to k−1.

At step 604 of the method 600, the decoding module 154 can determinewhether the particular selected bit position P_(k) has been found. Forexample, at step 658 of the method 650, the decoding module 154 candetermine whether the upper threshold does not equal the lowerthreshold. Here, equality is an indication that the selected bitposition P_(k) has been found.

If P_(k) has not been found, then the method 600 can proceed to step606, at which the decoding module 154 can compare the current indexvalue I to the value of the worth function ƒ(k, j), which can be, forexample, the choose function

$\begin{pmatrix}j \\k\end{pmatrix}{\quad,}$where j corresponds to the median sequence position count of the window.For example, the method 650 can proceed to step 660 in which thevariable j can be set to lower threshold +ceil ((upper threshold−lowerthreshold)/2). Here, should the average bit position between the upperand lower thresholds of the window not be an integer value, the function“ceil” or ceiling outputs the smallest integer that is not less than thevalue of the function's argument. In alternative implementations, a“floor” function, which outputs the largest integer that is not greaterthan the value of the function's argument, can be used in place of theceiling function. Subsequently, the method 650 can proceed to step 662,in which the decoding module 154 can determine whether the current indexvalue I is greater than or equal to the worth function ƒ(k, j). Asstated above, the worth function can, for example, be the choosefunction

$\begin{pmatrix}j \\k\end{pmatrix}{\quad.}$

At step 608, the decoding module 154 can reduce the window size based onthe comparison performed at step 606. For example, for worth functions,such as the choose function, that are employed to determine the index asdiscussed above with respect to step 252 of FIG. 2, the comparison withindex I can indicate whether the correct value of P_(k) is greater thanor less than the median value j. For example, if the current index valueI is greater than or equal to the value of the worth function ƒ(k, j),then P_(k) is greater than the median value j. Similarly, if the currentindex value I is less than the value of the worth function ƒ(k, j), thenP_(k) is less than the median value j. Thus, based on the comparison,the window can be reduced and in effect shifted towards the correctvalue of P_(k). For example, if at step 662 the decoding module 154determines that the current index value I is greater than or equal tothe value of the worth function ƒ(k, j), then the decoding module 154,at step 664, can set the lower threshold to the median value j, therebyshifting the lower boundary of the window towards the upper boundary. Ifat step 662 the decoding module 154 determines that the current indexvalue I is greater than or equal to the value of the worth function ƒ(k,j), then the decoding module 154, at step 668, can set the upperthreshold to j−1, thereby shifting the upper boundary of the windowtowards the lower boundary.

Thereafter, the comparison and the window size reduction can be iterateduntil P_(k) is found. For example, the method 600 can proceed to step604 and can determine whether P_(k) is found. If P_(k) is not found,then the comparison 606 and reduction 608 steps can be repeated. Forexample, in the method 650, subsequent to steps 664 and 668, thedecoding module 154 can proceed to step 658 in which the decoding module154 can determine whether the upper threshold is equal to the lowerthreshold. If not, then steps 660-664 or steps 660, 662 and 668 can beperformed until the upper threshold is equal to the lower threshold,which, as stated above, is an indication that the selected bit positionP_(k) has been found.

If the selected bit position has been found, as determined at step 604,then the method 600 can proceed to step 610 in which the value of P_(k)can be set. For example, in the method 650, if the decoding module 154determines that the upper threshold equals the lower threshold, then thedecoding module 154 can designate the bit position corresponding to thethresholds as P_(k) at step 670. In addition, the decoding module 154can decrement the current index by the worth function value of ƒ(k,P_(k)) and can decrement k by 1.

Thereafter, the method 600 can proceed to step 612, in which thedecoding module 154 can determine whether all selected bit positions(P_(k) for all k) have been found. For example, subsequent to step 670of the method 650, the decoding module 154 can proceed to step 654 anddetermine whether k is equal to 1. Should k equal 1, then all of theselected bit positions P_(k) have been found. When k reaches 1, theindex value is P_(k). Thus, when k is 1, the decoding module 154, atstep 672, can set P₁ to the current index value I and can therebydetermine that all P_(k) have been found. Thereafter, the methods 600and 650 may end. In the alternative, if, at step 612, the decodingmodule 154 determines that not all P_(k) have been found, then thedecoding module 154 can initiate the search process for the nextselected bit position P_(k) at step 612 and the method can proceed tostep 602 and be repeated until all selected bit positions P_(k) havebeen found. For example, if, at step 654, the decoding module 154determines that k is not 1, then the method 650 can proceed to step 656in which a new window is set. For example, in the new window, thedecoding module 154 can select a sequence or bit position count,P(k+1)−1, that precedes the bit position count, P(k+1), of a previouslyfound selected bit position as an upper-boundary (UT) of the window andcan select a sequence position count value, k−1, that precedes a valueof the selection occurrence count, k, of the selected bit position P_(k)that is currently sought as the lower-boundary (LT) of the window.Thereafter, the method can be repeated until all selected bit positionsP_(k) have been found.

It should be noted that because the calculation of the worth function,such as the choose function, can be computationally expensive for largevalues of k and P_(k), at least some of the values of the worth functioncan be stored a priori in a lookup table. This table can be used forboth indexing and reverse indexing procedures.

For example, the indexing module 114 of the transmitter system 110 canemploy the optional look-up table 115 to determine values of the worthfunction for summation operations and thereby determine the indices asdiscussed above with respect to step 252 of the method 200. Similarly,the receiver system 150 can employ the optional look-up table 155 todetermine values of the worth function in the various reverse indexingmethods discussed above with respect to methods 350-650. For example,the decoding module 154 of the receiver system 150 can employ theoptional look-up table 155 to determine values of the worth function forpurposes of comparison with the current index values at steps 356, 456,606 and 662. Accordingly, as discussed further herein below with respectto various exemplary implementations of lookup tables and worthfunctions, the indexing module 114 and/or the decoding module 154 canutilize a lookup table that cross-references a worth function ƒ(k, j)with corresponding worth function values to determine and employ thevalues for purposes of computing an index for a vector or determining acorresponding vector from the index.

In general, for counting all possible permutations of M ones and N-Mzeros,

$\left( {{{if}\mspace{14mu} M} < \frac{N}{2}} \right),$a table with NM entries can be employed. However, in accordance withvarious exemplary aspects, several different alternatives for a reducedsize table, and/or with reduced computation complexity, can be employedfor indexing and reverse indexing procedures.

For example, if a choose function is employed as the worth function,utilizing a table of

$\begin{pmatrix}j \\k\end{pmatrix}\quad$for 1≦k≦M and k≦j≦N−M+k as tables 115 and 155 enables a very fastcomputation. For example, using the table 155, the decoding module 154can complete a very fast computation of the sequence S by performing asearch operation in a sorted list

$\begin{pmatrix}N \\M\end{pmatrix},\begin{pmatrix}{N - 1} \\M\end{pmatrix},\ldots\mspace{14mu},\begin{pmatrix}M \\M\end{pmatrix}$to find the most significant nonzero or selected bit in the sequence S,and thereafter iteratively find the next nonzero or selected position tothe right of an already found position. Here, no multiplication isrequired. Further, by using the search methods discussed above, theupper bound on the number of comparisons to the index value is

${{{\log_{2}\left( {N - M} \right)} + {\sum\limits_{k = {M - 1}}^{1}\;{\log_{2}\left( {P_{k + 1} - k} \right)}}} \leq {M\mspace{11mu}{\log_{2}\left( {N - M} \right)}}},$which is a quite negligible computational complexity. The minimum sizeof the table is not more than (M−1)(N−M) integers.

In accordance with other exemplary aspects, the tables 115 and 155 canbe implemented as “reduced” lookup tables. Here, the indexing module 114or the decoding module 154 can compute choose function values from anentry in a lookup table by performing operations other than a worthfunction operation. For example, for the choose function

$\begin{pmatrix}j \\k\end{pmatrix},$the tables 115 and 155 can be configured to store only M−1 values

$\begin{pmatrix}{N - M + k} \\k\end{pmatrix},$where k=2, 3, . . . , M. For example, with respect to the reverseindexing procedures in which the sequence S is found from the receivedindex by the decoding module 154 discussed above, the search for eachnonzero or selected position P_(k), k=M, M−1, . . . , 1, can begin bycomparing

$\begin{pmatrix}{N - M + k} \\k\end{pmatrix}\quad$with 4. Thereafter, worth function values can be computed recursivelyfrom one or more entries in the reduced lookup table. For example, thenext value

$\begin{pmatrix}{N - M + k - 1} \\k\end{pmatrix}\quad$for comparison with I_(k) can be obtained by multiplying

${\begin{pmatrix}{N - M + k} \\k\end{pmatrix}\mspace{14mu}{by}\mspace{14mu} N} - \frac{M}{N - M + k}$and the next value

$\begin{pmatrix}{N - M + k - 2} \\k\end{pmatrix}\quad$is obtained by multiplying

${{\begin{pmatrix}{N - M + k - 1} \\k\end{pmatrix}\mspace{14mu}{by}\mspace{14mu} N} - M - \frac{1}{N - M + k - 1}},$and so on. In this case, the average operation employed to find eachposition P_(k) is

$\frac{N - M}{2}$integer multiplication operations and

$\frac{N - M}{2}$integer division operations. Thus, the total complexity is in the orderof (M−1)(N−M) multiplications and

$\frac{M\left( {N - M} \right)}{2}$comparison operations. Indeed, the complexity would usually be muchlower for indices structured as discussed above with regard to step 252,because after any nonzero or selected position is found, then thepossibilities for the next nonzero or selected position would be moreconstrained.

Alternatively, other functions can be employed as the worth function.For example, the worth function ƒdescribed above with respect to methods250-650 can be a function ƒ(k, l) which has the property, ƒ(k, l)≧ƒ(k,l−1)+ƒ(k−1, l−1), but is simpler to calculate than the binomial function

$\begin{pmatrix}{P(k)} \\k\end{pmatrix}.$One example of the function ƒ(k, l) is a function that can beimplemented by using a simplified operation, such as summation and shiftoperation, in place of a multiplication operation. If the function ƒ(k,l) is of relatively low complexity, then it can be directly computed.Otherwise, an alternative approach is to employ a function having a setof outputs that are relatively easier to store in a table, such astables 115 and/or 155. An example of such a function is one that outputsvalues that are in the form of a variable power of a prime number. Here,only the value of the variable power can be stored in tables 115 and/or155 instead of the actual value. Another example is a function thatoutputs values that are expressible in and that can be stored in theform of A×10^(n), where variable n can be stored directly in tables 115and/or 155 and A is a constant number that can be stored in one byte,i.e., 0≦A≦255.

In accordance with another approach, a function ƒ(k, l) can be designedthat is independent of the value of k and only depends on l. Here, thefunction can be employed as the worth function discussed above withrespect to methods 250 and 350. The function ƒ(k, l) in this scenarioshould have the following properties in order to produce a unique indexfor any permutation:

-   -   for all k, j and l, ƒ(k, l)=ƒ(j, l)=g(l);    -   for all        m _(j) , l _(j), 1≦j≦M:g(l ₁)+g(l ₂)+ . . . +g(l _(j))=g(m        ₁)+g(m ₂)+ . . . +g(m _(j))    -   if and only if {l₁, l₂, . . . , l_(j)}={m₁, m₂, . . . , m_(j)}.

The latter property indicates that if the set of {g(l), 1≦l≦N} isconsidered, the summation of every p elements of this set is unique for1≦p≦M. Accordingly, if this worth function is employed, the summation ofworth function values discussed above with respect to 252 is unique forany permutation of a selected element set (P_(k) for all k); in otherwords, each obtainable index value,

${{I(S)} = {\sum\limits_{k = 1}^{M}\;{f\left( {k,P_{k}} \right)}}},$from any potential sequence or vector S is unique. The function g(l) nowonly has N elements instead of MN elements and the indexing module 114or the decoding module 154 can utilize much less memory to store valuesg(l) v. l in tables 115 or 155, respectively, for purposes ofdetermining an index from a vector or determining a vector from anindex. However, the efficiency of the indexing procedure is lower thanthe alternative approach discussed above with respect to the function,ƒ(k, l) which has the property ƒ(k, l)≧ƒ(k, l−1)+ƒ(k−1, l−1). One way togenerate the function ƒ(k, l)=ƒ(j, l)=g(l) is to start with ƒ(0)=0 andƒ(1)=1 for a particular l. The element ƒ(k) is added as a strict upperbound to any partial summation of size not more than M elements of theset {ƒ(0), ƒ(1), . . . , ƒ(k−1)}. This construction ensures that thefunction retains the two properties described above.

Another approach can be employed as an alternative means for calculatingindex using the binomial function as the worth function. In particular,one or more values of the worth function can be deduced from one or moreentries in the lookup tables. The efficiency of this approach is similarto the approach utilizing the function with the property ƒ(k, l)≧ƒ(k,l−1)+ƒ(k−1, l−1). In other words, the resulting indices are exactlyconsecutive integer numbers for each permutation of selected positions.For example, the full table with MN entries with entries for

$\begin{pmatrix}P_{k} \\k\end{pmatrix}\quad$for 1≦k≦M and 1≦P_(k)≦N is considered, where each row of the table isfor a fixed value of k. Considering the fact that, for a binomialfunction ƒ(k, l), ƒ(k, l)=l for k=1, which indicates that there is noneed to store the first row of the table for k=1. To generate the table115 or 155 in this particular approach, an entry from every other columnof the table can be removed for each k; in other words, only the oddvalues of l are kept in the table. In particular, one or more values ofthe worth function can be deduced from one or more entries in the lookuptables. Using this reduced table, as opposed to performing the binomialfunction ƒ(k, l) operation directly, the binomial function ƒ(k, l) foreven values of l can be computed by performing only one addition ofvalues in the table entries and using the following identityƒ(k,l)=ƒ(k,l−1)+ƒ(k−1,l−1).For even values of l, rendering l−1 to be odd, ƒ(k, l) can be computedfrom ƒ(k, l−1) and ƒ(k−1, l−1), which are both entries of the reducedtable.

This reduction in the size of the table can be easily extended. Forexample, the table for ƒ(k, l) can include only the entries for which lis in the form of l=3a+1 for some integer value a. In other words, onlyentries for which the values of l is equal to 1 modulo 3 can be kept inthe table. Here, every value of ƒ(k, l) for l=3a+2 can be obtained in athe same manner described above by performing only one addition usingthe identity ƒ(k, l)=ƒ(k, l−1)+ƒ(k−1, l−1). The values of ƒ(k, l) withl=3a+3 can be similarly obtained by performing two additions and onebinary shift operation using the identityƒ(k,l)=ƒ(k,l−2)+2ƒ(k−1,l−2)+ƒ(k−2,l−2)

It should be noted that, here, multiplication by two is equivalent to abinary shift operation. In this example, the number of entries of thereduced table is only

$N = {\frac{{4N} - N}{3}.}$Thus, this example utilizes a reduced table with only N entries stored;the other values of ƒ(k, l) can be obtained without performing anymultiplication operations. The memory usage of this example is exactlyequal to the approach discussed above with regard to the function g(l),but the efficiency of the indexing is equivalent to the optimal indexingprocedure.

It should also be noted that it is possible to reduce the number ofentries of the table even further by employing the following identityfor the binomial function ƒ(k, l):ƒ(k,l)=ƒ(0,j)ƒ(k,l−j)+ƒ(1,j)ƒ(k−1,l−j)+ . . . +ƒ(j,j)ƒ(k−j,l−j)However, the above-referenced identity should be written in the formthat can be implemented with only addition and binary shift operationsin order to reduce the computation complexity. An example for j=3 isprovided herein below:T=ƒ(k−1,l−3)+ƒ(k−2,l−3)ƒ(k,l)=ƒ(k,l−3)+T+2T+ƒ(k−3,l−3).This calculation utilizes only four addition operations and one shiftoperation.

Having described preferred embodiments of systems and methods (which areintended to be illustrative and not limiting), it is noted thatmodifications and variations can be made by persons skilled in the artin light of the above teachings. It is therefore to be understood thatchanges may be made in the particular embodiments disclosed which arewithin the scope of the invention as outlined by the appended claims.Having thus described aspects of the invention, with the details andparticularity required by the patent laws, what is claimed and desiredprotected by Letters Patent is set forth in the appended claims.

1. A method for transmitting control information to permit theestablishment of a communication link comprising: determining an indexby computing ${I(S)} = {\sum\limits_{k = 1}^{M}{\begin{pmatrix}P_{k} \\k\end{pmatrix}{\quad,}}}$  wherein I(S) is the index, S is a binarysequence denoting selected subbands, M is the total number of selectedsubbands, k is a selection occurrence count of a selected subband andP_(k) is a subband position count in the binary sequence for a selectedsubband corresponding to k; and transmitting the index to enable theestablishment of a communication link on at least one selected subband.2. The method of claim 1, wherein the transmitting further comprisestransmitting an indication of the total number of selected subbands. 3.The method of claim 1, wherein selected subbands are denoted by ones. 4.The method of claim 1, wherein the bit positions of the binary sequenceare indexed from zero for the least significant position to N−1 for themost significant position, wherein N denotes the total number ofavailable subbands.
 5. The method of claim 1, wherein $\begin{pmatrix}P_{k} \\k\end{pmatrix}\quad$ is defined to be zero when P_(k) is less than k. 6.A method for transmitting control information to permit theestablishment of a communication link comprising: obtaining a binarysequence including at least one selected bit position representing atleast one subband selected from a plurality of available subbands;determining an index for the binary sequence as the summation of choosefunction values, wherein a particular choose function is defined by aselection occurrence count of a particular selected bit position that ischosen from a bit position count of the particular selected bitposition; and transmitting the index to enable the establishment of acommunication link on the at least one selected subband.
 7. The methodof claim 6, wherein the transmitting further comprises transmitting anindication of the total number of selected subbands.
 8. The method ofclaim 6, wherein selected subbands are denoted by ones.
 9. The method ofclaim 6, wherein the bit positions of the binary sequence are indexedfrom zero for the least significant position to N−1 for the mostsignificant position, wherein N denotes the total number of availablesubbands.
 10. The method of claim 6, wherein the choose function isdefined to be zero when the bit position count of the particularselected bit position is less than the selection occurrence count of theparticular selected bit position.
 11. A transmitter system comprising:an indexing module configured to determine an index for a binarysequence including at least one selected bit position representing atleast one subband selected from a plurality of available subbands,wherein the index is a summation of choose function values and whereinany particular choose function is defined by a selection occurrencecount of a particular selected bit position that is chosen from a bitposition count of the particular selected bit position; and atransmitter configured to transmit the index to enable the establishmentof a communication link on the at least one selected subband.
 12. Thesystem of claim 11, wherein the transmitter is further configured totransmit an indication of the total number of selected subbands.
 13. Thesystem of claim 11, wherein selected subbands are denoted by ones. 14.The system of claim 11, wherein the bit positions of the binary sequenceare indexed from zero for the least significant position to N−1 for themost significant position, wherein N denotes the total number ofavailable subbands.
 15. The system of claim 11, wherein the choosefunction is defined to be zero when the bit position count of theparticular selected bit position is less than the selection occurrencecount of the particular selected bit position.